The Quotient Rule: Finding the Derivative of f(x) = sec(x) Using the Quotient Rule of Differentiation

Dx {sec x}=?

To find the derivative of the function f(x) = sec(x), where sec(x) represents the secant function, we can use the quotient rule

To find the derivative of the function f(x) = sec(x), where sec(x) represents the secant function, we can use the quotient rule.

The quotient rule states that if we have a function f(x) = g(x) / h(x), where g(x) and h(x) are differentiable functions, then the derivative of f(x) with respect to x can be found using the formula:

f'(x) = (g'(x) * h(x) – g(x) * h'(x)) / (h(x))^2

In our case, g(x) = 1 and h(x) = cos(x).

Taking the derivatives of g(x) and h(x), we have:

g'(x) = 0 (since g(x) = 1 is a constant)
h'(x) = -sin(x) (by differentiating cos(x))

Now, substituting these values into the quotient rule formula, we get:

f'(x) = (0 * cos(x) – 1 * (-sin(x))) / (cos(x))^2
= sin(x) / (cos(x))^2

Therefore, the derivative of the function f(x) = sec(x) is f'(x) = sin(x) / (cos(x))^2.

Alternatively, we can rewrite sec(x) as 1 / cos(x) and use the chain rule to differentiate.

Chain rule states that if we have a composite function g(f(x)), then the derivative of g(f(x)) with respect to x is given by:

(dg/dx) = (dg/df) * (df/dx)

In our case, g(u) = 1 / u (where u = cos(x)).

Taking the derivative of g(u), we have:

(dg/du) = -1 / (u^2) (-1 by differentiating 1/u)

Now, taking the derivative of u = cos(x), we get:

(du/dx) = -sin(x) (by differentiating cos(x))

Substituting these values into the chain rule formula, we get:

(dg/dx) = (-1 / (cos(x))^2) * (-sin(x))
= sin(x) / (cos(x))^2

Again, we obtain f'(x) = sin(x) / (cos(x))^2.

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